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Thanks so much for your detailed response, Mitchell. That diagramme is especially helpful, and much better than the one I provided above!
We ended up using an adjustable workbench (pictured), with moveable 'dogs' to clamp the board from the sides, although we did end up needing one of us to put weight on the far end of the board to hold it down because of the amount of pressure we had to exert to get the holesaw fully through the board - we will definitely use clamps to vertically secure the remaining pieces!
With regard to the enamel paint, it turns out that the shelving is made from chipboard (not MDF), which will apparently expand and fall apart if we use a water-based enamel paint. Do you by any chance have some updated recommendations for sealing the exposed edges, now that you have the correct information? (Sorry for not realising that chipboard and MDF aren't the same thing!)
I'm glad to have introduced you to such a useful and cromulent word - I don't know how we ever got by without being able to embiggen things! Thank goodness for Jebediah Springfield. 
Okay, so the update is that our very good quality cordless AEG drill got a very hot motor and then started emitting smoke after getting maybe a third of the way through the chipboard. We switched to the cordless AEG hammer drill (pictured) which did NOT catch on fire, but still needed a *huge* amount of pressure to get all the way through - because we were only using an adjustable workbench with 'dogs' to clamp the board from the sides, we ended up needing one of us putting weight on the other end to hold it down.
It then started raining in the 30-odd minutes it took for us to wrangle the cut piece out of the holesaw, so we're going to resume the project tomorrow, using my stepdad's big fancy saw table and as many clamps as we can fit on there!! We'll also use the tip provided by TedBear below, cutting halfway through the chipboard from one side and then the other half from the other side, in the hope of the cutout not getting as firmly lodged inside the holesaw. I'll keep you all posted!
Very nicely done! @MsCharlie
Is that the big hole-saw seen in the background of your first photo above?
Any trouble getting it off the drill component afterwards?
I have found this 'pin' attachment really good with hole-saws.
Hello @MsCharlie
Thank you very much for the update on your cat tower project. It's good to see that you're making progress, just a friendly reminder that if your drill has a setting for drilling to set it to that function. Another way to seal the chipboard is to use an oil-based primer, it will be safe to use as it will not make it swell like water-based paints. Thank goodness for Jebediah Springfield!
Eric
Ah, the famous Eric L of DIY Cat Tower renown! I'm honoured to make your acquaintance. 
Updated update is that we bought a 152mm Sutton multipurpose holesaw that fits on the mandrill my stepdad already owned, and we've moved the project to his vertical drill press because using a hand drill is harder to control the RPM, whereas the drill press has a belt system to set exactly what speed is recommended for each job (the larger holesaws want 250 RPM, because of the torque). The drill press also guarantees even pressure from exactly above, so the hole won't be angled - this is especially useful when removing the cutout from the attachment, because if you cut even slightly wonky, the cutout piece will be wider than the holesaw attachment and REALLY jam itself in there!
I'm not super happy with the result from the multipurpose attachment, as it only has three quite fat cutting edges and has basically torn up the edges of the melamine - you can see the difference between the results in the images - the pile of holes is from the Sutton multipurpose attachment, and the other one is from the cheap brand. If I do this project again (I've already had requests from a bunch of my friends and family members!) I will get the Sutton attachment that says it's for metal, because it has lots and lots of little teeth, and I think that will cause the edges to be neater?
The sketchiness of the edges means that I will have to sand them (using fine paper wrapped around some dowel) and fill them with putty (packing it in and smoothing it with the dowel so I can kinda roll it and get a nice round finish) to make sure that the laminate doesn't peel off any further. I don't think a primer will do the job with the edges as ragged as they are this time around, but it sounds like a good option for future attempts with the other holesaw attachment - thanks for the tip!
Thanks for the updated update, @MsCharlie. You'd be correct in thinking that more teeth typically mean neater holes. You're also right that some timber filler and sanding will smooth out those edges; you'll then be able to prime and paint over the filler and inside of your cutouts.
I can't wait to see this finished!
Mitchell
Long-awaited update!
After filling the edges with putty and then sanding them, applying two coats of primer (sanding between), and several coats of high-gloss paint (lightly sanding with a very fine sandpaper between coats), the holes have been deemed acceptable for our fluffy tyrants to use as throughways. I've attached some photos for approval!
My partner and I both have ADHD, so we start a lot of projects at once, and we're only now assembling the shelves. Would it be useful for me to write up the whole project in a single entry and post it as a separate piece? I'm new here, and I don't know how helpful that would be.
Anyway, I'll post pics when we've assembled the shelves and the cats are steadfastly ignoring them, as is traditional!
Updated update, the shelves are now assembled. Thank you to everyone for your help, one of the jerks is already having a great time investigating!
Feel free to write up the project as a seperate piece @MsCharlie. I'm sure the community will be inspired by what you have created for your furry friends.
Thanks for sharing.
Jason
